Descent of uniform rationality from an algebraic closure

Let $K$ be a field, let $\bar{K}$ be an algebraic closure of $K$, and let $X_K\subset K^n$ be a nonsingular $K$-rational algebraic variety. Assume that $X_{\bar{K}}\subset\bar{K}^n$ is uniformly $\bar{K}$-rational. The uniform rationality descent conjecture asks whether

$$X_K\text{ is uniformly }K\text{-rational}.$$

The paper proves that, under these assumptions, uniform rationality holds after a finite-degree field extension of $K$, but the asserted descent to $K$ itself remains open. The rationality hypothesis is necessary because $\bar{K}$-rationality need not imply $K$-rationality.